The Parents' Review

A Monthly Magazine of Home-Training and Culture

"Education is an atmosphere, a discipline, a life."
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Teaching Arithmetic Part 2

by C.H. WilkinsonVolume 14, 1903, pgs. 654-663

[Note—this article had mathematical symbols and formatting that did
not transfer properly. LNL]

Teaching Arithmetic Part 2: Some weaknesses I have found and their remedy. (continued from page 578.)

I would now pass from general ideas to show how I would teach
arithmetic from its earliest stage. I begin with a child at six years
of age. The infant work below that age is such that there is nothing
vital as regards method in it. The latest developments of kindergarten,
which are well understood and well realized by infant teachers, need no
suggestion for their improvement. There are continuous advances being
made in this work, and they simply need the teacher to be on the alert
to gain the advantage of utilizing them. The chief principle adopted is
to teach the child to bring all his counts to ten. In adding he would
make ten of his first figures before proceeding to the next figures
beyond ten; and this he would do for every ten in succession.
Subtraction, as it is miscalled, is worked on the same method. At this
stage it is equally easy to break any object into ten parts, and give
the child to understand that each of those parts is a tenth; and that
there are ten tenths in a whole number as well as ten units in one ten.
It is likewise possible to teach him that a tenth may be represented in
two different ways, viz., 1 or 1/10. You teach the child that 1 is a
unit, and that "t" is one unit and one-tenth, and I think you might
teach that "i i" is one-tenth + one-tenth of one-tenth of a unit. This
must be shown to be of value before doing it. The addition and
subtraction and multiplication of decimals (or the two former at any
rate) can be taught at the same time as you teach ordinary addition and
subtraction; and so the child is at home at once. You can tell him also
that 1/10 means one whole number to be divided by 10 or into tenths.
You can put it that the top figure is to be divided by the bottom. You
could only do this at the age where you begin division.

But you can tell even at the earliest stages that one over two, as
"1/2" is the half, one over four, as "1/4," is the quarter, and so on,
of every simple fraction; and let them add halves together and see how
many they have, and how many whole ones they make, and so on, with
quarters and fifths and tenths, etc. At the stage where division comes
in you would say that 1/10 means one to be divided by 10. This is not
mathematically accurate, but it is accurate enough to get the
underlying idea. Of course in the case of 3/10 it would not always be
true that it was a tenth of three articles, or that 3 were to be
divided by 10 and one-tenth of each taken. In the cases of like values
it would be true. Take money, 3/10 of 13, or 1/10 of 3s is the same
thing; but 3/10 does not mean the 1/10 of 3s. Still the general
mathematical principle is here, that where you have a number on top it
has to be divided by the bottom in order to get its value and
relationship to the standard unit. That is the principle you want to
convey, and the distinction referred to is readily made clear to the
child at a more advanced age. In improper fractions it is more nearly
true. Thus 13/10 of an orange is an impossibility, for an orange only
has 10/10. But yet in calculations we get figures of this sort to deal
with as a short and ready way of getting to a desired result. What is
wanted at an early age is to get into the mind of the child of the idea
of how a fraction is represented and how it is easiest manipulated, and
later on he will regard it actually as what it is, viz., a known
portion of a unit. I sometimes go into a class of six-years old
children and say, what is the half of a half? The teacher regards me
for a moment with horror, and the children look blank. I know
beforehand they will do so. Then I put it in the concrete and say, if I
have an apple and divide it in half and give it to two of you, how much
each will the two have? They see that. Now suppose I take one half and
divide it into two and give to two of you, how much each will the two
have? One quarter they say right away. Then what is the half of a half?
Answer: 1/4. Then I ask, what did I do when I cut the half into two
equal parts? I get from them, "I halved it," or, "I divided it." Then
if I divide a half by two, what do I get? Answer: 1/4.

In Standard I., where division is going on, he drops to the fact that
division can be stated in fractional form. The teaching of both
fractions and decimals can be carried much further in this class.
Having utilized the fractional form in Standard II to the extent of
making him illustrate his mental arithmetic principles by it, and shown
him how to cancel out, I should have the foundation for algebra right
at the start of Standard III, or at least at 8 years of age for the
average child.

When in Standard II, let him take this sum: six dozen cost 6s.; what
will one article cost? If he tells me rightly, I want to know how he
has done it, and when he has done it in one way, I want the other way
also. First, ix dozen into 6s = one dozen for 1s. Multiply one dozen by
1s, and 1s. by 12. Get the reason for this. 12 singles into 12 pence =
1d. for one single. The second way is to reduce six dozen to single
ones = 6 x 12 =72. Reduce 6s. to pence = 6s. x 12 = 72d. Divide 72
pence by 72 singles; 72d. divided by 72 singles = 1d. Then I ask why one penny? Why not 1s.? Why divide
pence by single ones? Why not divide single ones by pence? It is
astonishing how little of this searching is done. Hence the mind of the
child is hazy, and his statements are indefinite. Now, let him state
the sum in fractional form. Next put 6 [British pound] for 6s. and let them give the
five ways in which the answer may by obtained and their reasons for
each step. In a little sum like this you have thus given them division,
multiplication, and several other principles incidentally, and while
they are doing something which to them is new, they are practising what
is old, and recapitulating without knowing it. By stating it in
fractional form, and showing how cancelling out can be done, you get
them on another stage. You can show them that, if you have (4x12)/6 it is the
same thing, whether you regard it as (xx)/12 = 8 or as (4xx)/8 = x/.4 = 8. [exact figures were mostly illegible]

After this at Standard III., at the start, I would show a child that
any hieroglyphic may represent value or number. I lead up to it by
saying that some people do sums in letters instead of figures. Taking
him from the known to the unknown, I say, "x dozen cost 6s., what is
the cost of one single article?"

State it fractionally, (6x12)/(3x12) Let some child give a figure for x and work it
out as above. Next take x dozen and y shillings = (?x12)/(xx12) Let two different
children give figures for y and x, and work it out as before. You need
to know from the child why y and x have to be multiplied by 12 each. If
you multiply one y by 12, how many y's do you get? You want to put all
sorts of questions as, Why not put x on the top, or why not put y at
the bottom? In this way the child learns that letters (any letters) may
have a value when they represent numbers, or money, or measure, etc.
You are familiarizing them with letters as a means of calculation,
while making yourself more positive that they know the principles of
their figures properly. This work can be done as blackboard work and
mental arithmetic. The children do the work in their heads and tell you
what to put down. A little done each day before the regular arithmetic
lesson will improve the written work of the boys themselves, and carry
them on very quickly, and make them confident and accurate in their
work. The converse is good in the higher classes. You can ask a boy to
square figures and work out problems mentally which would be hopeless
unless this style of training were made a feature. Take the formula (a
+ b), (a—b), and from it he can square quite larger figures. Take 372.
He would do it—

37 + 3 = 40
37—3 =34
(40 x 34) / (32) 9 = 37(2), viz., 1369.

Of course you do not tell him what formula to use. A boy of this sort
at this stage will select the right formula for the particular sum.
This is just one kind as an example. In this way you make one subject
help another, and get greater brilliancy in both. I went into a very
poor school looked at from the point of view of social standing and
funds. It was in the slums of a large midland town. In Standard III.,
when three moths' work had been done in that standard this was the work
they were doing when I went in. How many slates at Bd. Each can I buy
for £C: They work it on the lines I have indicated, viz., C x 240/B.
Boy gave a figure for C., viz., £25 Another boy gave 3d. for B. (25 x
240)/3 = (25 x 80)/1 = (100 x 80)/4 = (100 x 20)/1 = 2000/1 = 2000 slates. The third fractional equation was given to show that as
you could cancel out, so you could also multiply top and bottom by the
same figure without destroying the value of the fraction, or altering
it in any way. Also because at that school decimals were coming on, two
or three standards higher up. Therefore they prepared the boys somewhat
for them, by bringing 5's and 10's, and 20's and 25's to 100's. It also
showed them the value of decimals, inasmuch as they leaned that to
multiply by 10's or 100's was much easier. They could tell you why they
did everything, and it was quite easy to them and they liked it.

When teaching numeration I think few teachers show that in the same way
as you have 80 single articles you may have 80 items. If they are
dealing with 809, they will show that you have 8 hundreds and 9 single
units. They do not show that you have 80 tens and 9 units. When you
come to longer lines of figures, as 76,956, they show that they are 76
thousands and 9 hundreds and 5 tens, and 6 units. They never show that
there are 769 hundreds, or 7,695 tens, which is the very think the
children ought to know before starting long division. When teaching
multiplication, many children are never taught to know that it is
merely addition being done in the shortest way, and that 4 times 4 only
mean 4 + 4 + 4 + 4 added together. Nor in division are they led to
realize that division is only subtraction; the divisor being deducted
from the dividend a given number of times, and the result being that
there are so many of the divisor as are indicated by the quotient
contained in the dividend. 16 divided by 2 goes 8 times is only another
way of saying that 2 is contained 8 times or there are 8 twos in
sixteen. This is often not shown.

Even in numeration I have never known a teacher explain why 769 should
not mean 967. That is to say, that they do not say why tens are put to
the left of units and not to the right; nor why hundreds are put to the
left of the tens and units and not to the right of them. The child
should be shown that it is merely an arbitrary arrangement dogmatically
adhered to in order to avoid confusion, but that it could quite as well
have been arranged the other way. Of course the history of the question
could also be introduced. It should be pointed out to the child that on
the railway tickets of some lines the opposite order of arrangement
actually exists without causing any confusion to the company. In all
teaching of arithmetic the shortest method should be taught to the
child. He should know that the way he is taught is the shortest way,
and that the reason for doing it in that way rather than according to
some more lengthy method is to save time and unnecessary work. As is
the shortness of the day so must be the shortness of our work in order
to get as much done as possible. This is one important royal road to
success and highest usefulness in life. For this and many other
reasons, what is taught as subtraction should never be taught as
subtraction; nor should addition always taught as a column of figures.
Take subtraction. A boy does not in reality subtract anything at all He
finds the difference between two numbers. He should learn at the time
he begins his fractions, if not before that time, such signs as +, -,
./., etc., His so-called subtraction should be done simultaneously by
the use of minus signs and by placing the figures under one another. If
the figures are placed one under the other as in finding the difference
between 17 and 42, there is no reason why they should not be put with
17 above the 42 as well as with the 17 under the 42, thus 17 or 42. I
should put them sometimes one way and sometimes the other. The child at
present is taught almost that the sum is necessarily a subtraction sum because
the 17 is underneath. If he is taught that he has o find the difference
wherever it is he becomes much more alert and much less mechanical. In
any case the method of working should always unalterably be by adding
on to the smaller number enough to make the larger, and the child
should be clear that the amount thus added makes the difference between
the two numbers. This is what is known as the Italian method. The
teachers lose a lot of time and give themselves much extra work by not
adopting the Italian methods of subtraction and division. Here is a sum
I saw a clever master give his class.

The five lines of figures in bracket marked "A" had to
be added together and deducted from the top line B, column by
column as the boys came to them. The sum was on the
blackboard, and the boys stated what had to be done, and gave the
figure to be put down. Thus 6 + 5 + 2 + 4 + 9 + x in lines A = 2 in
line B, viz, 6 as shown in line C. Thus column after
column was done till you got the difference in line C of 15,923,386.
Its correctness is proved by adding the lines A to the line C, and
these will make the total in line B.

Just here is where saving in time of one description comes in, besides
altering the sum and making it more difficult. He drew the chalk
through the second and fourth columns o figures to the left of the
units, i.e., through the "tens" columns and the "1,000's" column, and
put Ls.d. over the top and added some fractions of 1d.

Again A lines were added and taken from line B, giving the difference
in line C between the other amounts. See the time saved just to run
chalk through and go on with the same sum and see the new notion the
boys get of the different values of figures, used in different
relationships. The advantages of subtraction done in this way are
manifold. First, it results in better method-training for the Italian
way of doing long division. I have found also that subtraction by
addition (or the Italian method) is quicker and more accurate as a rule.

Again, technically, it is more logical from an educational point of
view. A child is first taught addition; then under the old system he
would be taught something that appeared to be the antithesis of the
rule for addition. But under the system I advocate he goes on to
subtraction and does it under the guise of addition with which he has
just previously been made familiar. He simply adds on to a number
another number which shall be big enough to make the smaller number the
same as the larger. His first lesson had been to add several numbers on
the one to the other; or in other words to find the sum total of a
given set of numbers. Now he simply adds on to certain numbers enough
to make a given sum total. The process is intelligible to the child and
follows in natural sequence. Then again this process appeals to the
child because it is part of his daily life. If he is sent to purchase
goods the tradesman in giving change never subtracts. He adds on to the
cost of the goods the amount of change necessary to make up the amount
of coin the child tenders.

The chief reason of value other than these educational ones is that at
the present time the business books in smart houses are printed with
three columns on one page, Dr., Cr., Balance. The credits and debits
are worked on to the balance column right away. Suppose the balance at
your bankers to be a credit, all the credits for the day will be added
up (pence columns first of course) and the first column will be added
on to the pence in the balance column, and then in like manner the
shillings. The debits will be treated in a similar way, only they will
be deducted as you go along, the pence from the pence and so on, the
only figures appearing on the ledger being the entries and the
balances. No totals of credits or debits and no pencil entries. This
can only be done in one way, viz., by addition. The utilization of this
Italian method in doing long division I think invaluable. Take 679382
./. 348. Done in the ordinary way by long division it would be:—

The advantages are many. It is shorter. It is quicker. It is finer for
mental gymnastics. It tends to ensure more care, for if the child makes
a mistake he has to go through the whole sum again to find it; and so
it leads to greater accuracy. It is in harmony with previous methods of
teaching the ordinary elementary rules as subtraction. In long division
usually the greatest number of mistakes occur in the subtraction. By
this method multiplication and subtraction are done simultaneously.

With the long division the advance of knowledge in decimals is readily
extended and increased.

Suppose 214,705 is to be divided by 203.

1,057
203)214,7,0,5
1170
1220
134

In this instance I have put the quotient over the
dividend instead of at the side. I think it makes it clearer to
the child. I should do this in ordinary long division. The sum was done first as an ordinary long division sum. Then a
decimal point was put in the middle and the boys asked where the point
should be in the quotient. It was shifted from point to point in the dividend and the boys shifted
it readily in the quotient. The reason was given for the 1 going over
the 4 and the 0 over the 7 and so on.

There is much I could refer to on this subject, but space and time
suggest my confining to one other matter, and that is the nature of the
sums set. I have said I like the principle of the Government B scheme,
because of its problems and mode of progression, which suits well the
training of the child. Some books have excellent little problems in
them. I think, however, there is a field for more definite purpose in
setting sums. I would like all arithmetic to represent some facts in
relation to other subjects which should unconsciously impress those
facts on the mind of the child.

Take history, for example, I would set a sum in this form. The battle
of Waterloo was fought in 1815, Wellington had for his army so many
each of English, Dutch, and Belgian troops. How many had he altogether?
Napoleon had such a number. How many men were in the field under the
two leaders? Blucher brought so many more to Wellington's aid. How many
did this make? Then deal with the losses by the flight of the Belgians,
the deaths on each side, the prisoners taken from time to time, and you
get your sums. Then say the total on each side were so many, how many
more would Wellington have needed if the French had had 10 per cent
more than they had, and so on for proportion sum. Divide the force into
divisions under their generals. Give one general x number, another one
y number, and another one z number of men. Tell them that x stands for
so many, and y for another quantity, and z for another number, thus—x/47,000 y/47,000 z/47,000. Ask them how many altogether. So at this
early stage you get them familiar with the use of letters. Use actual
numbers as far as history reveals them.

Take another example in geography. A steamer takes 125 hours to go from
Liverpool to Philadelphia, and she steams 25 miles an hour. How many
miles are the ports distant? The sum should be so set as to the time
taken and the rate per hour, as to give an actual distance for the
answer. You might even take the time of the record steamer to date. My
example is illustrative and the figures are not actual in any sense.

You could take all the various ports and take the distances they are
apart in various parts of the would. Then take the various rates of
sailing and steaming; and work out from these data the length of time
it would take the different vessels. Problems set on these lines would
be most valuable. Mountains and their height, rivers and their length,
towns and their distances apart, can all be utilized for like purposes.
Physics give scope for lovely, simple, and interesting arithmetic
problems, and in the more advanced arithmetic, astronomy could to some
extent be impressed into our service. I have not touched on many
important features of arithmetic dealing with factors, etc. I think we
all realize the importance of attention to these. The sole idea of this
paper is to suggest some newer and more satisfactory method of dealing
with all rules and at all stages, so that the teaching may be more
regularly and gradually progressive—more thorough in the ground-work
and more usefully co-ordinated with other subjects. What I have
suggested in regard to the elementary principles of the most elementary
educational work should, I think, possibly apply, as far as the
principles are concerned, to more advance work. In any case, I am
certain that the advanced work of our rising generation would be
greatly improved, and the children would be more successful if the
elementary work were done with greater attention to detail and with a
more minute application to the development of underlying principles.